System Of Equations Research Articles

Compatibility of strains and the three-fold differentiability of the displacement field

The problem of the necessary class of smoothness of solutions to quasi-static problems of deformable solid mechanics in terms of displacements was discussed. It is shown that in order for the equations of compatibility of deformations to become identities when displacements are substituted in them, the existence of some third mixed derivatives of displacements is required. A counterexample for a linearly elastic compressible isotropic elastic medium was given. In this counterexample, the displacement field, being a doubly differentiable solution to the boundary value problem for the system of Lame equations in the entire domain, is not a solution to the displacement problem at all points in this domain.

Classical-quantum simulation of non-equilibrium Marshak waves

In the radiation hydrodynamic simulations used to design inertial confinement fusion (ICF) and pulsed power experiments, nonlinear radiation diffusion tends to dominate CPU time. This raises the interesting question of whether a quantum algorithm can be found for nonlinear radiation diffusion which provides a quantum speedup. Recently, such a quantum algorithm was introduced based on a quantum algorithm for solving systems of nonlinear partial differential equations (PDEs) which provides a quadratic quantum speedup. Here, we apply this quantum PDE (QPDE) algorithm to the problem of a non-equilibrium Marshak wave propagating through a cold, semi-infinite, optically thick target, where the radiation and matter fields are not assumed to be in local thermodynamic equilibrium. The dynamics is governed by a coupled pair of nonlinear PDEs which are solved using the QPDE algorithm, as well as two standard PDE solvers: (i) Python's py-pde solver; and (ii) the KULL ICF simulation code developed at Lawrence-Livermore National Laboratory. We compare the simulation results obtained using the QPDE algorithm and the standard PDE solvers and find excellent agreement.

Non-fragile sampled-data observers resisting measured noise for a class of nonlinear systems

In this paper, we study non-fragile observers for a class of nonlinear systems with sampled outputs and unknown measurement noise. After an output filtering transformation, an extended nonlinear system is constructed. The sampled outputs with unknown measurement noise are transformed into the state equation of the extended system. Then, an extended non-fragile observer is designed to estimate the states of the extended system, and the unknown measurement noise is estimated by a first-order estimator. On the basis of a time-varying matrix inequality (TMI), it is proven that the observer system can track the extended system well even if there are observer gain perturbations and measured noise. Finally, a nonlinear dynamic model is used to verify the feasibility and validity of the proposed methods.

Theory of thin elastic plate: history and current state of the problem

The article is an analytical review and is devoted to the theory of thin, isotropic elastic plates. There are basic relations of the theory based on the kinematic hypothesis confirming that the tangential displacements are distributed linearly along the thickness of the plate and its deflection does not depend on the normal coordinate. As a result, a system of equations of the sixth order with respect to two potential functions – the penetrating potential, which determines the plate deflection, and the boundary potential which makes it possible to set three boundary conditions on the plate edge and eliminate the known contradiction of Kirchhoff's plate theory was obtained. Problems that have no correct solution in the framework of Kirchhoff's theory – cylindrical bending of a plate with a free edge, bending of a rectangular plate with non-classical hinge fixing, torsion of a square plate by moments distributed along the contour, and bending of a plate by a rigid die – are considered. In conclusion, a brief historical review of the papers devoted to the theory of plate bending was presented.

Evaluating the mixed convection flow of Casson fluid from the semi-infinite vertical plate with radiation absorption effect

This study examines a steady laminar Casson fluid flow induced by a semi-infinite vertical plate under the impact of the Darcy-Forchheimer relation and thermal radiation. The features of mixed convection, cross-diffusion, radiation absorption, heat generation, chemical reactions and viscous dissipation are also considered to explain the transport phenomenon. The resultant system of equations, concerned with the problem under consideration, is transformed into a group of non-linear ordinary differential equations (ODEs) by means of similarity variables. The bvp4c method, an instrument popular for its numerical accomplishments, is utilized to solve this problem. The effect of flow parameters on heat transfer, concentration and velocity is evaluated via diagrams. To validate our code, we have compared the present outcomes to the prevenient literature, and stable consent has been detected. Moreover, the friction coefficient Cfx, Nusselt number Nux, and Sherwood number Shx are also computed to assess velocity gradient, efficiency of heat transfer and mass transfer process, respec-tively.

On a contact problem for a homogeneous plane with a finite crack under friction

An exact solution to the contact problem of indentation of an absolutely rigid punch with a straight base, taking into account friction, into one of the edges of a finite crack located in a homogeneous elastic plane was derived. It is assumed that shear contact stresses are directly proportional to normal contact pressure. In this case, it is assumed that the friction coefficient is directly proportional to the coordinates of the contacting points of the contacting surfaces. The governing system of equations for the problem was derived in the form of the heterogeneous Riemann problem for two functions with variable coefficients and its closed solution is constructed in quadratures. Simple formulas for contact stresses and the normal dislocation component of displacements of crack edge points were obtained. The patterns of changes in contact stresses and crack opening depending on the maximum value of the friction coefficient have been studied.

Paired piecewise linear regression model with fixed nodes

In this paper we develop a method for constructing a paired regression model in the class of piecewise linear continuous functions with fixed nodes. The concept of linear division of the correlation field, its partial sections and its nodes is introduced into consideration. Using the linear division of the correlation field, a class of piecewise linear functions with fixed nodes is determined. The linear division is revealed during the visual analysis of the correlation field. Using the least squares method, a system of linear equations is compiled to find point estimates of the parameters of the approximating function. With the exception of two unknown angular coefficients (unknown) this system is reduced to a tridiagonal system of equations, the unknowns of which are the nodal values of the approximating function. The tridiagonal system is solved by the run-through method. An algorithm was constructed to demonstrate the operation of the developed method. Its initial data is arrays of the corresponding values of the explanatory and dependent variables, as well as an array of numbers of the right ends of the intervals defining the nodes. An array of fixed nodes is constructed from an array of values of the explanatory variable and an array of numbers of the right ends of the intervals. Next, an array of point estimates of the parameters of the approximating function is constructed. This algorithm is implemented in Python in the form of user-defined functions.

PARASIAN OVER PARISIAN, HOW MUCH EARLIER SHOULD ONE EXERCISE?

In this paper, an integral equation approach used for pricing American-style Parisian options is extended to pricing Parasian options, after overcoming an additional difficulty of losing the “reset” feature of the latter in addition to still dealing with the high nonlinearity associated with American-style exercises. More specifically, compared with an American Parisian option, the accumulative feature of the “tracking clock” time results in a pair of coupled three-dimensional (3D) Partial Differential Equation (PDE) systems instead of coupled two-dimensional (2D) and 3D PDE systems. Upon successfully establishing the couple integral equation systems, we have successfully computed the option price and the optimal exercise price and compared the results with those of its Parisian counterpart. Through the comparison, we can directly observe how the accumulativeness of the “tracking clock” time affects the option and the optimal exercise prices and provide some meaningful financial explanations.

Numerical Study of Complex Heat Transfer and Aerodynamics in a Tube Furnace with a Flat Fuel Combustion Mode

The object of the study is a tubular furnace for steam reforming of natural gas. The channel walls are formed by a refractory lining and a tubular screen with a known temperature. The chamber volume is filled with a selectively radiating, absorbing and weakly scattering medium of gaseous fuel combustion products. The research method is based on the numerical solution of a system of two-dimensional integro-differential equations of radiative gas dynamics, closed by a two-parameter model of turbulent convection. It is shown that when burners are located on the side walls of the radiation chamber, a significant restructuring of the heat flux distribution occurs. A decrease in the chamber width is accompanied by an increase in both radiant and convective heat fluxes to the tubular screen.

Evaluating the risk of malaria transmission within the Central African Republic with the goal of stabilising and eliminating the infection

In this paper, we formulate a multi-compartmental mathematical model for humans and mosquitoes. We construct the system of differential equations for an SEITVR for the human compartment and an SEI for the mosquito compartment. We investigate the outbreak of malaria and its effect on the Central African Republic. The analysis of the compartmental model is carried out using stability analysis and Routh Hurwitz Criterion technique is used to indicate the major impact of the model and to improve the model through minor modifications in the transformation of disease in the population. Our model exhibits two equilibrium points, disease free equilibrium points and endemic equilibrium points. The next generation matrix is used to determine the basic reproduction number R 0. A new compartmental model was framed and estimated the malaria spread after 2023 in the Central African Republic, which is the novelty of this research. Our main motivation is to make the Central African Republic a malaria free country. A Numerical example are provided to validate our results for both the disease free state and endemic state of each model. We believe that this investigation will be more effective in reducing malaria infection and stop spreading.

New aspects of the development of Kantorovich’s one-product dynamic model of replacement of production funds

In the context of the development of scientific and technological progress and the growth of the capital stock, an important task is modeling and optimization of the terms of operation of production funds. The development of technologies and automation of production lead to a reduction in the workforce employed in production. In the article, for the previously developed single-product dynamic model of the replacement of production assets, taking into account the inertial properties of the funds being introduced, the case of a reduction in labor resources under the condition of an increase in output and capital investment is investigated. A solution was obtained that allows to determine the optimal strategy for the withdrawal of obsolete funds and the introduction of the new ones in case of decrease in labor resources. As the optimization criterion the principle of differential optimization is used. The theorem of equivalence of a trajectory satisfying the principle of differential optimization, the obtained solution is given. A system of functional differential equations is presented, which should be satisfied by variable models both in terms of growth and reduction of labor resources. Numerical methods are used to solve such a system. The variants of the system development are considered under various assumptions regarding changes in the total amount of labor resources. The results of numerical implementation of the model are presented.

A New Operational Matrices Based-Technique for Fractional Integro Reaction-Diffusion Equation Involving Spatiotemporal Variable-Order Derivative

In this article, a novel generalized model of the nonlinear fractional integro-reaction-diffusion equation (NFIRDE) with spatiotemporal variable-order (SVO) is introduced, where the variable order derivatives are equipped with the Atangana–Baleanu–Caputo (ABC) sense. This model represents a great generalization of a significant type of NFIRDE and their applications. Moreover, A novel, efficient and fully spectral shifted Legendre tau technique is developed to solve the proposed model. Despite the difficulty of applying this mechanism to solve this type of equations, due to the presence of nonlinear terms and the SVO functions that appear in the traditional differential and integral operational matrices. We deduce some new operational matrices that play the fundamental role in facilitating the implementation of the tau method. These operational matrices represent the SVO ABC-derivative, the integro term within the model, as well as the vector multiplications with the space-time Kronecker product. As a result, the proposed model is restructured into a system of nonlinear algebraic equations, which simplifying the solving process. We illustrate our method’s effectiveness and validity with numerical examples with both smooth and non-smooth solutions. Our findings show that the proposed tau method delivers accurate results and exhibits non-local properties.

Numerical Estimation of Bending in Holographic Volume Gratings by Means of RCWA and Deep Learning

In this paper, we introduce a novel approach to model bending phenomena on holographic volume gratings based on Rigorous Coupled Wave Analysis (RCWA), in which the bending as a phase in the dielectric permittivity expansion is introduced, and the Shooting Method (SM) is employed to solve the resulting system of equations. Further validation of our model is conducted by comparing its predictions to those obtained from reference Finite-Difference Time-Domain (FDTD) simulations and Coupled Wave Theory (CWT, referring to Kubota’s model that includes the bending phenomenon). Furthermore, we propose a methodology for estimating the bending from the diffraction efficiency curves in transmission volume gratings based on deep learning models, with a subsequent study of their accuracy and applicability.

A study of self-similar vector fields in bianchi type III spacetime via Rif tree approach

Abstract In this study, we use the Rif tree approach to explore self-similar vector fields in Bianchi type III spacetime. This work adopt a computer-based method to transform symmetry equations into an involutive form that is simplified and divides the integration problem into multiple cases, each represented as a tree structure. In some cases, the metric functions are subject to particular constraints. These conditions allow one to solve the system of equations governing self-similar symmetries and provide explicit formulations for the metrics and their corresponding self-similar vector fields. This approach is novel in that it covers not only the analysis of the direct integration strategy but also some metrics that are practically relevant. For a detailed investigation of the created models, stability and physical importance are also considered.

A Novel Approach for Time-Local Fractional Solutions of Certain Nonlinear Partial Differential Equations in Fractal Dimension

Time-local fractional approaches for nonlinear partial differential equations in fractal dimensions are essential for capturing the complex, irregular behaviors found in fractal systems. In this paper, a new modification of the local fractional Laplace variational iteration method (MLFLVIM) for obtaining analytical approximate solutions to the fractional gas dynamics equation, fractional Stefan equation, and fractional Newell-Whitehead-Segel equation within the context of fractal time space is presented. The proposed method (MLFLVIM) elegantly combines the local fractional Laplace transform (LFLT) with modified variational iteration method. Specifically, we first apply the (LFLT) to the given local fractional PDEs, yielding a transformed system of equations. We then apply modified variational iteration to this system. Finally, we use the inverse of (LFLT) to obtain the desired solution. To demonstrate the effectiveness of this approach, we implement it on three numerical physical problems. The results show that the (MLFLVIM) can successfully handle these nonlinear LFPDEs and provide accurate analytical approximation solutions.

Numerical Simulation Study of Thermal Performance in Hot Water Storage Tanks with External and Internal Heat Exchangers

This paper presents a numerical analysis of two hot water storage tank configurations—one equipped with an external heat exchanger (Tank-1) and the other with an internal heat exchanger (Tank-2). The objective is to evaluate and compare their thermal performance during charging and discharging processes. The numerical model is developed by solving a system of ordinary differential equations using the 4th-order Runge–Kutta method, implemented in the Python programming language. The results indicate that Tank-1 demonstrated a higher charging efficiency of 94.6%, achieving full charge in approximately 2 h and 20 min. In comparison, Tank-2 required 3 h and 47 min to reach full charge, with a charging efficiency of 85.9%. During discharge, both configurations exhibited similar behavior, with an efficiency of 13.63% over approximately 33 min. The analysis showed that the external heat exchanger configuration led to more effective thermal stratification, supported by the Richardson number analysis, which indicated a significant effect of buoyancy during charging. This design advantage makes Tank-1 particularly suitable for applications requiring rapid heating and minimal heat loss, such as in cold climates or intermittent demand systems. The numerical model demonstrated reliable predictive accuracy, achieving an RMSE of 6.1% for the charging process and 6.8% for the discharging process, thereby validating the model’s reliability. These findings highlight the superior performance of the external heat exchanger configuration for fast and efficient energy storage, particularly for applications in cold climates.

Numerical solutions and stability analysis of hybrid Casson nanofluid flow with MHD and heat transfer effects

AbstractThis research addresses the complex dynamics of hybrid Casson nanoliquids in flow and heat transfer applications, focusing on the interaction between fluid dynamics and thermal phenomena in the presence of magnetohydrodynamics (MHD), viscous dissipation, and Joule heating. The motivation behind this study stems from the need to enhance the efficiency of heat transfer processes in various engineering applications, such as cooling systems and electronic devices, where hybrid nanofluids can offer superior thermal performance compared to conventional fluids. The novelty of this work lies in its comprehensive numerical investigation of a hybrid Casson nanoliquid over a moving permeable surface, incorporating a detailed analysis of MHD effects, viscous dissipation, and Joule heating. Using the Tiwari and Das model to formulate the governing equations, the study employs similarity transformations to convert these equations into a system of ordinary differential equations (ODEs). MATLAB is then used to derive numerical solutions, with a stability analysis ensuring the physical dependability of these solutions. Key findings reveal that the temperature distribution of the hybrid nanoliquid shows a positive correlation with both Prandtl and Hartmann numbers. Additionally, a positive relationship between temperature and the Eckert number is observed. These insights offer valuable guidance for engineers aiming to optimize heat transfer processes using hybrid nanofluids, highlighting their potential for improved thermal management in practical applications.

The influence of lymphatic vessels on nanoparticle distribution and heat transfer within tissue

AbstractThis study analytically investigates the dynamics of nanoparticle transport within a three‐dimensional porous cylinder simulating a lymphatic vessel, without external heat sources. The governing equations and boundary conditions are transformed to yield a system of ordinary differential equations, which are solved numerically using MATLAB built‐in function, bvp4c. Key parameters are visually examined and physically interpreted in relation to temperature, velocity, concentration, and Nusselt number profiles. The study reveals that the distribution of temperature and Nusselt number are maximized by increasing the heat transfer coefficient, whereas NP concentration is increased by decreasing it. Furthermore, the Brownian motion parameter enhances both heat transmission and NP concentration. It is also observed that simpler extravasation into lymphatics decreases tissue nanoparticle levels and heat conduction. Ultimately, optimal intra‐lymphatic nanoparticle distribution pathways are achieved by specifically varying heat transfer and interstitial mass flux patterns. By simulating biological barriers and lymphatic drainage, this model enhances our understanding of the underlying transport mechanisms controlling nanoparticle mobilization.

Numerical Study of the Influence of Boundary Conditions on Calculations of the Dynamics of Polydisperse Gas Suspension

The work numerically simulates the flow of a polydisperse gas suspension in a channel. The carrier medium was described as a viscous, compressible, heat-conducting gas. The mathematical model implemented a continuum technique for the dynamics of multiphase media, taking into account the interaction of the carrier medium and the dispersed phase. For each component of the mixture, a complete hydrodynamic system of equations of motion for the carrier phase and dispersed phase fractions was solved. The dispersed phase consisted of particles with different sizes of dispersed inclusions. For the carrier medium, homogeneous Dirichlet boundary conditions were specified on the side surfaces of the channel. For fractions of the dispersed phase, boundary conditions for slippage. The influence of the boundary conditions of the flow of the carrier medium on the dynamics of gas suspension fractions has been revealed.

APPLICATION OF OPTIMIZATION METHOD IN MULTIDIMENSIONAL SPACE TO THE PROBLEM OF DETERMINING FULLERENE CONCENTRATION IN FULLERENE-CONTAINING MIXTURE

The work proposes a method for the quantitative processing of experimental data from ultraviolet spectroscopy of substituted fullerene mixtures, taking into account errors in the experimental data. This task can be written as a system of Vierordt equations (a system of linear equations). The coefficients of the system are the extinction coefficients of individual substituted fullerenes, which are difficult to measure accurately. Therefore, the mathematical model of the problem contains significant errors that affect the reliability and stability of the solution. To minimize the influence of errors, the work [1] proposes a method, which involves reducing the considered system of linear equations to a linear programming problem. This study generalizes the work [1] and proposes to use linear programming problems to find interval values of correction factors for parameters containing errors. Since decreasing the discrepancy reduces the solution error only for well-conditioned coefficient matrices, the values of correction coefficients that minimize the condition number of the coefficient matrix are found in the obtained intervals. The found correction values allowed to significantly improve the conditionality of the system coefficient matrix and obtain a more accurate solution. By using the refined extinction coefficients, the distribution of molar fractions of fullerene fragments and fullerene-containing compounds in the mixture of products formed by the attachment of cyanoisopropyl radicals was obtained, as well as the distribution of molar fractions of fullerene fragments of different degrees of substitution for samples of fullerene-containing polymethyl methacrylate and the distribution of molar fractions of fullerene fragments of different degrees of substitution for samples of fullerene-containing polystyrene. The results are correct and consistent with the theory, both for low-molecular-weight mixtures and for fullerene-containing polymers.